Problem 23 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{(x+7)(x+8)}{(x+8)(x+7)} $$

Step-by-Step Solution

Answer

Question: Simplify the following rational expression to its lowest terms: %% \(\frac{(x+7)(x+8)}{(x+8)(x+7)}\) Answer: 1

Step 1: Identify the factors of the numerator and the denominator

In this given rational expression, we can see that the numerator and the denominator are already factored into two binomials each: \((x+7)(x+8)\) for the numerator and \((x+8)(x+7)\) for the denominator.

Step 2: Cancel out common factors

Since both the numerator and the denominator have common factors \((x+7)\) and \((x+8)\), we can cancel them out: $$ \frac{(x+7)(x+8)}{(x+8)(x+7)} = \frac{(x+7)\cancel{(x+8)}}{\cancel{(x+8)}(x+7)} $$

Step 3: Simplify the expression

After canceling out the common factors, we are left with: $$ \frac{(x+7)}{(x+7)} $$ Since both the numerator and the denominator are the same, the final simplified rational expression is: $$ 1 $$ The rational expression has been simplified to its lowest terms, which is 1.

Key Concepts

Factoring Polynomials

Understanding how to factor polynomials is a key step in handling rational expressions. Factoring is essentially the process of breaking down a polynomial into simpler pieces, called factors, that when multiplied together will result in the original polynomial.
In the exercise given, both the numerator and the denominator were already presented as factored forms, namely \((x + 7)(x + 8)\). This is quite convenient because it allows you to quickly identify whether these expressions can be simplified further.
Factoring polynomials generally involves these steps:

• Look for common factors in all terms and take them out.
• Identify patterns such as the difference of squares: a² - b² = (a - b)(a + b).
• Apply specific factorization formulas or techniques like factoring trinomials.

The goal is to express the polynomial as a product of simpler polynomial factors. Practicing this skill is fundamental because it is frequently used in simplifying algebraic fractions and rational expressions.

Simplifying Expressions

Simplifying expressions, especially in algebra, often comes down to reducing the complexity by canceling out common factors.
In the example provided, the expression \( \frac{(x+7)(x+8)}{(x+8)(x+7)} \) was simplified by canceling out the common binomial factors in the numerator and the denominator. This left you with a straightforward expression of \( \frac{(x+7)}{(x+7)} \), which equals 1 since anything divided by itself is 1 (assuming it's not zero).
Here are key points to consider when simplifying expressions:

• First, factorize both the numerator and the denominator if they aren't already.
• Look for and cancel out any common factors, as they do not change the value of the expression.
• Check your simplified expression to ensure it can't be reduced further.

Effective simplification may require practice, but once mastered, it is an essential tool to handle more complicated algebraic equations with ease.

Algebraic Fractions

Algebraic fractions, or rational expressions, are fractions where the numerator and/or the denominator is a polynomial. Working with these types of expressions involves similar principles as numerical fractions, but it includes additional steps such as factoring and cancelling out polynomials.
Here’s a simple checklist for dealing with algebraic fractions:

• Simplify the rational expression by factoring polynomials in both numerator and denominator.
• Cancel out any common factors between the numerator and the denominator.
• Ensure that the denominator is not zero, which would make the fraction undefined.

The exercise provided is a classic example of an algebraic fraction that can be simplified by recognizing common factors. It's crucial to become comfortable with these steps since they form the backbone of manipulating more complex rational expressions. Remember, practice makes perfect when it comes to managing algebraic fractions effectively!